Question

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{3}+3 x^{2}+3 x+2\)

Short answer

The function \(y=x^{3}+3 x^{2}+3 x+2\) has a point of inflection and a relative maximum at the point \((-1,1)\).

Step-by-step solution

Find the first derivative

Differentiate the given function \(y = x^{3}+3x^{2}+3x+2\) with respect to \(x\). This will result in the first derivative \(y' = 3x^{2}+6x+3\). The critical points of the function are obtained by setting the first derivative equal to zero and solving for \(x\). Hence, \(3x^{2}+6x+3 = 0\).

Solve for critical points

Solving the equation \(3x^{2}+6x+3 = 0\) yields the critical points. You can factor the quadratic equation to find the roots, which are \(x = -1\).

Determine the critical points

Substitute the critical points into the original function to get the corresponding \(y\) -values. Hence, when \(x = -1\), \(y = -1 + 3 - 3 + 2 = 1\). So, the relative minimum or maximum occurs at the point \((-1,1)\).

Compute the second derivative

To classify the critical points determined in the previous step, compute the second derivative of the original function. This results in \(y'' = 6x+6\).

Classify the critical points

Use the critical points to determine whether the second derivative is positive or negative. If \(y''(-1)>0\), the critical point is a local minimum. If \(y''(-1)<0\), the critical point is a local maximum. Calculate \(y''(-1)\) to get -6, so \((-1,1)\) is a relative maximum.

Find the inflection points

The points of inflection of the graph can be determined by setting the second derivative equal to zero and solving for \(x\). Hence, \(6x+6 = 0\), which solves to give \(x = -1\). Substituting \(x = -1\) into the original function gives the point of inflection as \((-1,1)\).

Sketch the graph

Having determined the relative extrema and points of inflection of the function, proceed to sketch the graph. The graph has a point of inflection at \((-1,1)\) and a relative maximum at \((-1,1)\).